Porous solids equations

D is replaced by DeB for porous solids. Equation (2.369) agrees in terms of its form with (2.53) for heat conduction in a rod. In this case the temperature profile was calculated with the boundary conditions of constant temperature at the beginning of the rod and vanishing heat flow at the end of the rod. In the current problem these correspond to the boundary conditions (2.370) and (2.371). The solution of (2.369) to (2.371) therefore corresponds to the relationship (2.59) found earlier for heat conduction in rods. The solution is... 

The surface area of the pore openings makes up only a fraction p of the outer surface area of a porous solid (Equation 2.1-25). 

The adsorption branch of isotherms for porous solids has been variously modeled. Again, the DR equation (Eq. XVII-75) and related forms have been used [186,194]. With respect to desorption, the variety of shapes of loops of the closed variety that may be observed in practice is illustrated in Fig. XVII-29 (see also Refs. 195 and 197). 

The physical adsorption of gases by non-porous solids, in the vast majority of cases, gives rise to a Type II isotherm. From the Type II isotherm of a given gas on a particular solid it is possible in principle to derive a value of the monolayer capacity of the solid, which in turn can be used to calculate the specific surface of the solid. The monolayer capacity is defined as the amount of adsorbate which can be accommodated in a completely filled, single molecular layer—a monolayer—on the surface of unit mass (1 g) of the solid. It is related to the specific surface area A, the surface area of 1 g of the solid, by the simple equation... 

Any interpretation of the Type I isotherm must account for the fact that the uptake does not increase continuously as in the Type II isotherm, but comes to a limiting value manifested in the plateau BC (Fig. 4.1). According to the earlier, classical view, this limit exists because the pores are so narrow that they cannot accommodate more than a single molecular layer on their walls the plateau thus corresponds to the completion of the monolayer. The shape of the isotherm was explained in terms of the Langmuir model, even though this had initially been set up for an open surface, i.e. a non-porous solid. The Type I isotherm was therefore assumed to conform to the Langmuir equation already referred to, viz. 

In writing the present book our aim has been to give a critical exposition of the use of adsorption data for the evaluation of the surface area and the pore size distribution of finely divided and porous solids. The major part of the book is devoted to the Brunauer-Emmett-Teller (BET) method for the determination of specific surface, and the use of the Kelvin equation for the calculation of pore size distribution but due attention has also been given to other well known methods for the estimation of surface area from adsorption measurements, viz. those based on adsorption from solution, on heat of immersion, on chemisorption, and on the application of the Gibbs adsorption equation to gaseous adsorption. 

It would be difficult to over-estimate the extent to which the BET method has contributed to the development of those branches of physical chemistry such as heterogeneous catalysis, adsorption or particle size estimation, which involve finely divided or porous solids in all of these fields the BET surface area is a household phrase. But it is perhaps the very breadth of its scope which has led to a somewhat uncritical application of the method as a kind of infallible yardstick, and to a lack of appreciation of the nature of its basic assumptions or of the circumstances under which it may, or may not, be expected to yield a reliable result. This is particularly true of those solids which contain very fine pores and give rise to Langmuir-type isotherms, for the BET procedure may then give quite erroneous values for the surface area. If the pores are rather larger—tens to hundreds of Angstroms in width—the pore size distribution may be calculated from the adsorption isotherm of a vapour with the aid of the Kelvin equation, and within recent years a number of detailed procedures for carrying out the calculation have been put forward but all too often the limitations on the validity of the results, and the difficulty of interpretation in terms of the actual solid, tend to be insufficiently stressed or even entirely overlooked. And in the time-honoured method for the estimation of surface area from measurements of adsorption from solution, the complications introduced by... 

Static and dynamic property The uses of these foams or porous solids are used in a variety of applications such as energy absorbers in addition to buoyant products. Properties of these materials such as a compressive constitutive law or equation of state is needed in the calculation of the dynamic response of the material to suddenly applied loads. Static testing to provide such data is appealing because of its simplicity, however, the importance of rate effects cannot be determined by this one method alone. Therefore, additional but numerically limited elevated strain-rate tests must be run for this purpose. 

The volume of stationary phase with which the solutes in a mixture can interact (Vs in equation (11)) will depend on the physical nature of the stationary phase or support. If the stationary phase is a porous solid, and the sizes of the pores are commensurate with the molecular diameter of the sample components, then the stationary phase becomes... 

Streaming Potential When the solution is forced through the porous solid under the effect of an external pressure P, the character of liquid motion in the cylindrical pores will be different from that in electroosmotic transport. Since the external pressure acts uniformly on the full pore cross section, the velocity of the liquid will be highest in the center of the pore, and it will gradually decrease with decreasing distance from the pore walls (Fig. 31.5). The velocity distribution across the pore is quantitatively described by the Poiseuille equation. 

The study of how fluids interact with porous solids is itself an important area of research [6], The introduction of wall forces and the competition between fluid-fluid and fluid-wall forces, leads to interesting surface-driven phase changes, and the departure of the physical behavior of a fluid from the normal equation of state is often profound [6-9]. Studies of gas-liquid phase equilibria in restricted geometries provide information on finite-size effects and surface forces, as well as the thermodynamic behavior of constrained fluids (i.e., shifts in phase coexistence curves). Furthermore, improved understanding of changes in phase transitions and associated critical points in confined systems allow for material science studies of pore structure variables, such as pore size, surface area/chemistry and connectivity [6, 23-25],... 

Vapor sorption onto porous solids differs from vapor uptake onto the surfaces of flat materials in that a vapor (in the case of interest, water) will condense to a liquid in a pore structure at a vapor pressure, Pt, below the vapor pressure, P°, where condensation occurs on flat surfaces. This is generally attributed to the increased attractive forces between adsorbate molecules that occur as surfaces become highly curved, such as in a pore or capillary. This phenomenon is referred to as capillary condensation and is described by the Kelvin equation [19] ... 

The specific surface area. 1 of a porous solid that is formed of the monodisperse particles (solid phase) of volume I, and surface area. , is equal to the multiplication of the surface area of one particle by the number of particles Am in unit of mass. Using Equation 9.50,... 

Newman, A. B. Trans. Am. Inst. Chem. Eng. 27 (1931) 203. The drying of porous solids diffusion and surface emission equations. 

The advantage of equation 17.14 is that it may be fitted to all known shapes of adsorption isotherm. In 1938, a classification of isotherms was proposed which consisted of the five shapes shown in Figure 17.5 which is taken from the work of Brunauer et alSu Only gas-solid systems provide examples of all the shapes, and not all occur frequently. It is not possible to predict the shape of an isotherm for a given system, although it has been observed that some shapes are often associated with a particular adsorbent or adsorbate properties. Charcoal, with pores just a few molecules in diameter, almost always gives a Type I isotherm. A non-porous solid is likely to give a Type II isotherm. If the cohesive forces between adsorbate molecules are greater than the adhesive forces between adsorbate and adsorbent, a Type V isotherm is likely to be obtained for a porous adsorbent and a Type III isotherm for a non-porous one. 

Wheeler [16] proposed that the mean radius, r, and length, L, of pores in a catalyst pellet (of, for that matter, a porous solid reactant) are determined in such a way that the sum of the surface areas of all the pores constituting the honeycomb of pores is equal to the BET (Brunauer, Emmett and Teller [17]) surface area and that the sum of the pore volume is equed to the experimental pore volume. If represents the external surface area of the porous particle (e.g. as determined for cracking catalysts be sedimentation [18]) and there are n pores per unit external area, the pore volume contained by nSx cylindrically shaped pores is nSx nr L. The total extent of the experimentally measured pore volume will be equal to the product of the pellet volume, Vp, the pellet density, Pp, and the specific pore volume, v. Equating the experimental pore volume to the pore volume of the model... 

If it is assumed that the gaseous reactant is consumed by reaction with the porous solid at a rate i A and that transport of reactant within the solid is by a diffusive mechanism characterised by an effective diffusivity Dg, then the conservation equation for a reactant A in an isothermal... 

The assumption usually made is that the ratio Fu /Sbet has the same value at a given relative pressure independent of the solid. A plot therefore of t versus P/Pq should give the same curve for any non-porous solid (see Fig. 8.6). In fact, plots of the number of adsorbed layers versus P/Pq show some discrepancies which for the analysis of large pores is not significant. Therefore, the Halsey equation can be used for the statistical thickness in that application. However, for micropore analysis, a statistical thickness must be taken from a t versus P/Pq curve that has approximately the same BET C value as the test sample. The unavailability of t versus P/Pq plots on numerous surfaces with various C values would make the MP method of passing interest were it not for the fact that t can be calculated from equation (8.36). This implies that surface area can be accurately measured on microporous samples. Brunauer points out that in most instances the BET equation does correctly measure the micropore surface area. 

The particle and bulk densities are commonly used in mass balance equations, since the mass and the external volume of the particles are involved. On the other hand, the hydraulic density should be preferably used in hydrodynamic calculations, because buoyancy forces are involved, and so the total mass of the particle should be taken into account, including the fluid in the open pores. It is obvious that the particle density is equal to the skeletal and hydrodynamic density in the case of nonporous particles. Moreover, in the case of a porous solid in a gas-solid system, the gas density is much lower than the particle density, and tlius... 

In an apparatus based on Figure 6.16b, the volume of mercury forced into the pores of the solid can be measured as a function of the applied pressure. Equation (86) shows that higher pressures are required for smaller pores. Therefore incremental increases in p will result in the filling of pores of progressively smaller radii. The volume V that has intruded a porous solid at a pressure p gives the cumulative volume of all pores larger than the size associated with p. Since plots of V versus p give information about the cumulative pore distribution, it is the derivative of such data that measures the increment in pore volume associated with an increment in Rc. Written as a formula, dV/dp oc dV/(—dRc) since Fincreases as R( decreases. 

As with Equation (86), Rc can be eliminated from Equation (94) by calibration with a liquid for which 8 = 0. In that case Equation (94) can be used to determine 8 for powders or porous solids. Alternatively, a value of 8 may be assumed, and Rc can be evaluated from the rate of mercury intrusion. Equation (94) is called the Washburn equation. 

Adsorption hysteresis is often associated with porous solids, so we must examine porosity for an understanding of the origin of this effect. As a first approximation, we may imagine a pore to be a cylindrical capillary of radius r. As just noted, r will be very small. The surface of any liquid condensed in this capillary will be described by a radius of curvature related to r. According to the Laplace equation (Equation (6.29)), the pressure difference across a curved interface increases as the radius of curvature decreases. This means that vapor will condense... 

The overall reaction rate may be affected by diffusion if the mass transfer from the bulk fluid to the coal surface and/or the pore diffusion steps are relatively slow. I have made tentative estimates of both these diffusion effects. The following equation can be written (2) for the rate of oxygen uptake of a porous solid such as coal, assuming the oxygen to be consumed in an irreversible first-order reaction. 

Diffusion and Mass Transfer During Leaching. Rates of extraction from individual panicles arc difficult to assess because il is impossible to define the shapes of the pores or channels through which mass transfer has to take place. However, the nature of the diffusional process in a porous solid could be illustrated by considering the diffusion of solute through a pore. This is described mathematically hy the diffusion equation, the... 

For gas-solid heterogeneous reactions particle size and average pore diameter will influence the reaction rate per unit mass of solid when internal diffusion is a significant factor in determining the rate. The actual mode of transport within the porous structure will depend largely on the pore diameter and the pressure within the reactor. Before developing equations which will enable us to predict reaction rates in porous solids, a brief consideration of transport in pores is pertinent. 

This equation (which corresponds with equation 17.53 in Volume 2, Chapter 17), however, cannot be directly applied to the majority of porous solids since they are not well represented by a collection of straight cylindrical capillaries. Everett(I0)... 

McGreavy and THORNTON(23) have developed an alternative approach to the problem of identifying such regions of unique and multiple solutions in packed bed reactors. Recognising that the resistance to heat transfer is probably due to a thin gas film surrounding the particle, but that the resistance to mass transfer is within the porous solid, they solved the mass and heat balance equations for a pellet with modified boundary conditions. Thus the heat balance for the pellet represented by equation 3.24 was replaced by ... 

In one of the most common sensors, the non-porous solid electrolyte layers takes the form of a crucible closed at one end so that air used as a reference gas can be introduced into the interior of the crucible while the outside of the crucible is exposed to the exhaust gas. A schematic drawing and E versus the air-fuel ratio curve for this sensor are shown in Figure 1. E is the electromotive force (EMF) between the two electrodes in accordance with the Nemst equation ... 

The theory of Brunauer, Emmett and Teller167 is an extension of the Langmuir treatment to allow for multilayer adsorption on non-porous solid surfaces. The BET equation is derived by balancing the rates of evaporation and condensation for the various adsorbed molecular layers, and is based on the simplifying assumption that a characteristic heat of adsorption A Hi applies to the first monolayer, while the heat of liquefaction, AHL, of the vapour in question applies to adsorption in the second and subsequent molecular layers. The equation is usually written in the form... 

The BET model can also be applied to a situation which might be applicable to porous solids. If adsorption is limited to n molecular layers (where n is related to the pore size), the equation... 

Although the BET equation is open to a great deal of criticism, because of the simplified adsorption model upon which it is based, it nevertheless fits many experimental multilayer adsorption isotherms particularly well at pressures between about 0.05 p0 and 0.35 pQ (within which range the monolayer capacity is usually reached). However, with porous solids (for which adsorption hysteresis is characteristic), or when point B on the isotherm (Figure 5.5) is not very well defined, the validity of values of Vm calculated using the BET equation is doubtful. 

Use the Kelvin equation to calculate the pore radius which corresponds to capillary condensation of nitrogen at 77 K and a relative pressure of 0.5. Allow for multilayer adsorption on the pore wall by taking the thickness of the adsorbed layer on a non-porous solid as 0.65 nm at this relative pressure. List the assumptions upon which this calculation is based. For nitrogen at 77 K, the surface tension is 8.85 mN m-1 and the molar volume is 34.7 cm3 mol-1. 

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